ch6_arccomrm

Chapter 6 - Momentum and Collisions

Image from: http://www.dhss.mo.gov/InterventionMICA/Images/momentum.jpg

Section One - Momentum and Impulse
Momentum is a vector quantity described by the product of an object's mass times its velocity:
$\displaystyle \vec{p}=m\vec{v}$
If an object's momentum is known, its kinetic energy can be found as follows:
$\displaystyle KE=\frac{p^2}{2m}$
The change in the momentum of an object is equal to the impulse delivered to the object. The impulse is equal to the constant net external force acting on the object times the time over which the force acts:
$\displaystyle \Delta\vec{p}=\vec{F}\Delta t$
Any force acting on an object will cause an impulse, including frictional and gravitational forces
In one dimension the slope of a graph of the momentum of an object vs. time is the net external force acitng on the object. The area under a graph of the net external force acting on an object vs. time is the total change in momentum of that object.
Momentum and impulse are measured in:
$\displaystyle kg\frac{m}{s}$ -----there is no special name for that unit

Section Two - Conservation of Momentum

The law of conservation of momentum is the total momentum of all objects interacting with one another remains constant regardless of the nature of the force between the objects:


$\displaystyle \Sigma\vec{p}_i=\Sigma\vec{p}_f$
which is:
$\displaystyle m_1\vec{v}_1_i+m_2\vec{v}_2_i=m_1\vec{v}_1_f+m_2\vec{v}_2_f$



This relationship is true for all interactions between isolated objects. The change in momentum of the first object is equal to and opposite the change in momentum of the second object.


An Explanation of the Conservation of momentum:
-To get to the conservation of momentum problem, you must consider Newton's Third law:
Newton's Third law leads to conservation of momentum. It tells us that the change in momentum of the first object is equal to and opposite the change in momentum of the second object.
-The force on m1 is equal to and opposite the force on m2 (f1= - f2), the impulse on m1 is equal to and opposite the impulse on m2; therefore the change in momentum of m1 is equal to and opposite the change in momentum m2.

from http://webpages.uah.edu/~wilderd/momentum.jpg

Image from: http://webpages.uah.edu/~wilderd/momentum.jpg

Section Three - Elastic and Inelastic Collisions

There are three types of collisions: elastic, inelastic, and perfectly inelastic.

Elastic collision: A collision in which the total momentum and the total kinetic energy remain constant; the two objects return to their original shapes with no change in total kinetic energy or momentum; no energy is lost

Image from: http://physicslearning.colorado.edu/PiraHome/ResourceCD/ResourceImages/PhysicsDrawings/Elastic_Collision.gif

• $\displaystyle \Sigma\vec{p}_i=\Sigma\vec{p}_f$
• $\displaystyle \Sigma KE_i>\Sigma KE_f$

Perfectly Inelastic collision: A collision in which two objects stick together and move with a common velocity after colliding; the final mass is equal to the combined masses of the two objects.

inage from: http://physicslearning.colorado.edu/PiraHome/ResourceCD/ResourceImages/PhysicsDrawings/Inelastic_Collision.gif

• $\displaystyle m_1\vec{v}_{1i}+m_2\vec{v}_{2i}=m_1+\vec{v}_{1f}+m_2\vec{v}_{2f}$
• $\displaystyle \frac{1}{2}m_1v_{1i}^2+\frac{1}{2}m_2v_{2i}^2>\frac{1}{2}m_1v_{1f}^2+\frac{1}{2}m_2v_{2f}^2$

Inelastic collsion: the total kinetic energy does not remain the constant when the objects collide and stick together; the two objects in the collision are deformed and lose some kinetic energy. The decrease in kinetic energy can be calculated using the formula for kinetic energy from Chapter 5.

Image from: http://www.co.washington.or.us/sheriff/media/photos/h2_wreck.jpg

General Review Major Equations:

• Momentum
  • $\displaystyle \vec{p}=m\vec{v}$
• Kenetic Energy (Given Momentum)
  • $\displaystyle KE={p^2}{2m}$
• Impulse
  • $\displaystyle \Delta\vec{p}=\vec{F}\Delta t$
• Conservation of Momentum (General)
  • $\displaystyle \Sigma\vec{p}_i=\Sigma\vec{p}_f$
• Conservation of Momentum (Two Objects)
  • $\displaystyle m_1\vec{v}_1_i+m_2\vec{v}_2_i=m_1\vec{v}_1_f+m_2\vec{v}_2_f$
• Change in Momentum
  • $\displaystyle \Delta\vec{p}_1=-\Delta\vec{p}_2$
• Elastic Collisions (General)
  • $\displaystyle \Sigma\vec{p}_1=\Sigma\vec{p}_f$
  • $\displaystyle \Sigma KE_i=\Sigma KE_f$
• Elastic Collisions (Two Objects)
  • $\displaystyle m_1\vec{v}_1_i+m_2\vec{v}_2_i=m_1\vec{v}_1_f+m_2\vec{v}_2_f$
  • $\displaystyle \frac{1}{2}m_1v_1^2_i+\frac{1}{2}m_2v_2^2_i=\frac{1}{2}m_1v_1^2_f+\frac{1}{2}m_2v_2^2_f$
• Inelastic Collisions (General)
  • $\displaystyle \Sigma\vec{p}_i=\Sigma\vec{p}_f$
  • $\displaystyle \Sigma KE_i>\Sigma KE_f$
• Inelastic Collisions (Two Objects)
  • $\displaystyle m_1\vec{v}_{1i}+m_2\vec{v}_{2i}=m_1+\vec{v}_{1f}+m_2\vec{v}_{2f}$
  • $\displaystyle \frac{1}{2}m_1v_{1i}^2+\frac{1}{2}m_2v_{2i}^2>\frac{1}{2}m_1v_{1f}^2+\frac{1}{2}m_2v_{2f}^2$

Information from Holt Physics textbook